179 research outputs found

    Exponential Asymptotics in a Singular Limit for nn-Level Scattering Systems

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    The singular limit \eps\ra 0 of the SS-matrix associated with the equation i\eps d\psi(t)/dt=H(t)\psi(t) is considered, where the analytic generator H(t)\in M_n(\C) is such that its spectrum is real and non-degenerate for all tRt\in\R. Sufficient conditions allowing to compute asymptotic formulas for the exponentially small off-diagonal elements of the SS-matrix as \eps\ra 0 are explicited and a wide class of generators for which these conditions are verified is defined. These generators are obtained by means of generators whose spectrum exhibits eigenvalue crossings which are perturbed in such a way that these crossings turn to avoided crossings. The exponentially small asymptotic formulas which are derived are shown to be valid up to exponentially small relative error, by means of a joint application of the complex WKB method together with superasymptotic renormalization. The application of these results to the study of quantum adiabatic transitions in the time dependent Schr\"odinger equation and of the semiclassical scattering properties of the multichannel stationary Schr\"odinger equation closes this paper. The results presented here are a generalization to nn-level systems, n2n\geq 2, of results previously known for 22-level systems only.Comment: 35 pages, Late

    Density of States and Thouless Formula for Random Unitary Band Matrices

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    We study the density of states measure for some class of random unitary band matrices and prove a Thouless formula relating it to the associated Lyapunov exponent. This class of random matrices appears in the study of the dynamical stability of certain quantum systems and can also be considered as a unitary version of the Anderson model. We further determine the support of the density of states measure and provide a condition ensuring it possesses an analytic density

    An Adiabatic Theorem for Singularly Perturbed Hamiltonians

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    The adiabatic approximation in quantum mechanics is considered in the case where the self-adjoint hamiltonian H0(t)H_0(t), satisfying the usual spectral gap assumption in this context, is perturbed by a term of the form ϵH1(t)\epsilon H_1(t). Here ϵ0\epsilon \to 0 is the adiabaticity parameter and H1(t)H_1(t) is a self-adjoint operator defined on a smaller domain than the domain of H0(t)H_0(t). Thus the total hamiltonian H0(t)+ϵH1(t)H_0(t)+\epsilon H_1(t) does not necessarily satisfy the gap assumption, ϵ>0\forall \epsilon >0. It is shown that an adiabatic theorem can be proven in this situation under reasonnable hypotheses. The problem considered can also be viewed as the study of a time-dependent system coupled to a time-dependent perturbation, in the limit of large coupling constant.Comment: 17 pages, LaTe

    Spectral Properties of Non-Unitary Band Matrices

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    We consider families of random non-unitary contraction operators defined as deformations of CMV matrices which appear naturally in the study of random quantum walks on trees or lattices. We establish several deterministic and almost sure results about the location and nature of the spectrum of such non-normal operators as a function of their parameters. We relate these results to the analysis of certain random quantum walks, the dynamics of which can be studied by means of iterates of such random non-unitary contraction operators.Comment: updated version, to appear in Annales Henri Poincar

    Fractal Weyl Law for Open Chaotic Maps

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    This contribution summarizes our work with M.Zworski on open quantum open chaoticmaps (math-ph/0505034). For a simple chaotic scattering system (the open quantum baker's map), we compute the "long-living resonances" in the semiclassical r\'{e}gime, and show that they satisfy a fractal Weyl law. We can prove this fractal law in the case of a modified model.Comment: Contribution to the Proceedings of the conference QMath9, Mathematical Physics of Quantum Mechanics, September 12th-16th 2004, Giens, Franc

    Dynamical Localization of Quantum Walks in Random Environments

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    The dynamics of a one dimensional quantum walker on the lattice with two internal degrees of freedom, the coin states, is considered. The discrete time unitary dynamics is determined by the repeated action of a coin operator in U(2) on the internal degrees of freedom followed by a one step shift to the right or left, conditioned on the state of the coin. For a fixed coin operator, the dynamics is known to be ballistic. We prove that when the coin operator depends on the position of the walker and is given by a certain i.i.d. random process, the phenomenon of Anderson localization takes place in its dynamical form. When the coin operator depends on the time variable only and is determined by an i.i.d. random process, the averaged motion is known to be diffusive and we compute the diffusion constants for all moments of the position

    Spectral Properties of Quantum Walks on Rooted Binary Trees

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    We define coined Quantum Walks on the infinite rooted binary tree given by unitary operators U(C)U(C) on an associated infinite dimensional Hilbert space, depending on a unitary coin matrix CU(3)C\in U(3), and study their spectral properties. For circulant unitary coin matrices CC, we derive an equation for the Carath\'eodory function associated to the spectral measure of a cyclic vector for U(C)U(C). This allows us to show that for all circulant unitary coin matrices, the spectrum of the Quantum Walk has no singular continuous component. Furthermore, for coin matrices CC which are orthogonal circulant matrices, we show that the spectrum of the Quantum Walk is absolutely continuous, except for four coin matrices for which the spectrum of U(C)U(C) is pure point
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